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Abstract We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever$$M = M_1 \ast M_2$$is a tracial free product von Neumann algebra and$$u_1 \in \mathscr U(M_1)$$,$$u_2 \in \mathscr U(M_2)$$are Haar unitaries, the relative commutants$$\{u_1\}' \cap M^{\mathcal U}$$and$$\{u_2\}' \cap M^{\mathcal U}$$are freely independent in the ultraproduct$$M^{\mathcal U}$$. Our proof relies on Mei–Ricard’s results [MR16] regarding$$\operatorname {L}^p$$-boundedness (for all$$1 < p < +\infty $$) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan–Ioana–Kunnawalkam Elayavalli’s recent construction [CIKE22] to provide the first example of a$$\mathrm {II_1}$$factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.
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Conformal blocks for Galois covers of algebraic curves
We study the spaces of twisted conformal blocks attached to a$$\Gamma$$-curve$$\Sigma$$with marked$$\Gamma$$-orbits and an action of$$\Gamma$$on a simple Lie algebra$$\mathfrak {g}$$, where$$\Gamma$$is a finite group. We prove that if$$\Gamma$$stabilizes a Borel subalgebra of$$\mathfrak {g}$$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$$\Gamma$$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$$\mathscr {G}$$be the parahoric Bruhat–Tits group scheme on the quotient curve$$\Sigma /\Gamma$$obtained via the$$\Gamma$$-invariance of Weil restriction associated to$$\Sigma$$and the simply connected simple algebraic group$$G$$with Lie algebra$$\mathfrak {g}$$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$$\mathscr {G}$$-torsors on$$\Sigma /\Gamma$$when the level$$c$$is divisible by$$|\Gamma |$$(establishing a conjecture due to Pappas and Rapoport).
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- Award ID(s):
- 2001365
- PAR ID:
- 10519991
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 159
- Issue:
- 10
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 2191 to 2259
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X23007418
